Journal of Function Spaces
Volume 2016 (2016), Article ID 6814326, 6 pages
http://dx.doi.org/10.1155/2016/6814326

Research Article

The Harmonic Bloch and Besov Spaces on the Real Unit Ball by an Oscillation

Xi Fu,¹ Zhiyao Xu,² and Xiaoyou Liu³

¹Department of Mathematics, Shaoxing University, Shaoxing, Zhejiang 312000, China
²Department of Mathematics, Jiangxi Vocational College of Industry and Engineering, Pingxiang, Jiangxi 337055, China
³School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, China

Received 25 November 2015; Accepted 28 February 2016

Academic Editor: Ruhan Zhao

Copyright © 2016 Xi Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be the real unit ball in and . Given a multi-index of nonnegative integers with , we set the quantity where and . In terms of it, we characterize harmonic Bloch and Besov spaces on the real unit ball. This generalizes the main results of Yoneda, 2002, into real harmonic setting.

1. Introduction

Let be the real unit ball in with , where is the normalized volume measure on and is the normalized surface measure on the unit sphere . We denote the class of all harmonic functions on the unit ball by . For , denotes the gradient of . Given a multi-index of nonnegative integers, we use the notations and

For each , the harmonic -Bloch space consists of all functions such that

and the little -Bloch space consists of the functions such that

The harmonic Besov space is the space of all functions in for which

where and is the invariant measure on .

Let be a continuous function in . If there exists a constant such that

for any , then we say that satisfies weighted Lipschitz condition. By means of it, Ren and Kähler [1] obtained the following.

Theorem A. Let . Then if and only if it satisfies the weighted Lipschitz condition.

Theorem B. Let and . Then if and only if

Let be the open unit disk in the complex plane and let be a continuous complex-value function in . Denote by

where is the hyperbolic disc with center and radius , an integer, , and .

In [2], Yoneda characterized harmonic Bloch and Besov spaces in in terms of as follows.

Theorem C. Let be a complex-value harmonic function in . Fix an integer and a pair of real numbers , such that . Then if and only if is bounded.

Theorem D. Let be a complex-value harmonic function in . Fix an integer and a pair of real numbers , such that . Then for each , if and only if

where .

See [3–5] for various characterizations of the Bloch, little Bloch, and Besov spaces in the unit ball of .

The main purpose of this paper is to give some characterizations for the spaces , , and in the real unit ball along Yoneda’s direction. In Section 2, we collect some known results that will be needed in the proof of our results. Our main results and their proofs are presented in Sections 3 and 4.

Throughout this paper, constants are denoted by ; they are positive and may differ from one occurrence to the other. The notation means there is a positive constant such that .

2. Preliminaries

We will be using the same notation in [1, 6]: we write in polar coordinates by and . For any , let

Then the symmetry lemma in [7] shows that

For any , denote by the Möbius transformation in . It is an involution of such that and , which is of the form

By simple computations, we have

For any and , we define the pseudo-hyperbolic ball with center and radius as

Clearly, .

Lemma 1 (see [1, Lemma 2.1]). Let and . Then

where denotes the volume of .

The following is a characterization of the space (resp., ) which is proved in [8].

Lemma 2. Let and be a positive integer. Then (resp., ) if and only if

for all multi-index with .

As an application of Lemma 2, we can obtain the following.

Lemma 3. Let . Then if and only if for each ,

Proof. Fixing a point and letting

with , we have

for each . By Lemma 2, we see that .
For the converse, we assume that . Let ; then for each

It follows from [9] that there exists such that

This implies that

So the result follows.

Combining Theorem A and Lemma 3, we extend [2, Corollary 2.4] into the real harmonic setting as follows.

Corollary 4. Let . Then for , if and only if

where for all .

In the following, we give an example which shows that Corollary 4 does not hold for .

Example 5. Let ; then . By a simple computation we have

3. Results and Discussions

3.1. Harmonic Bloch Spaces

In this section, we give some characterizations of the spaces which can be viewed as the generalizations of Yoneda’s results into the real unit ball of .

For a continuous function in and , we write

By using the notation, we characterize as follows.

Theorem 6. Let ,   be an integer and . Then if and only if

for all multi-index with , where .

Proof. First we prove the sufficiency. Let . Then for each multi-index ,  . For , it follows from [1] that

Fixing and replacing by , we have

By Lemma 1, we can deduce that

Since is a constant, we see that

for any multi-index with . Hence Lemma 2 yields that .
Now we prove the necessity. Let . Then for each multi-index with , we have

By Lemma 1 we infer that there exists such that and

Thus,

So the proof is complete.

Theorem 7. Let , be an integer and . Then if and only if

for all multi-index with , where .

Proof. Sufficiency: assume that (33) holds. Then for any , there exists such that

whenever . It follows from an argument similar to that in proof of Theorem 6 that we have

whenever . Hence

from which we see that .
Necessity: for , let . By Lemma 1 and proof of Theorem 6, we see that, for each multi-index with ,

for all and . So

First letting and then letting , we obtain the desired result.

In the following, by removing the restriction in Theorem 6, we obtain the following.

Theorem 8. Let , , and . Then if and only if

Proof. We only need to prove the necessity since the proof of sufficiency is similar to that in proof of Theorem 6. Assume that . For , ,

which gives

where the last integral converges since . Thus

Similarly, we can prove the following.

Theorem 9. Let , , and . Then if and only if

3.2. Harmonic Besov Space

In order to prove our next result, we need the following lemma.

Lemma 10. Let . Then if and only if

for all multi-index with .

Proof. This follows from [10, Theorem 3.7] by letting .

Lemma 11. Let and . Then there exist constants ,   such that

Proof. By Cauchy’s estimates and Lemma 1, for each , we have

for some .

Now, we come to state and prove the result for harmonic Besov spaces.

Theorem 12. Let , be an integer and . Then if and only if

for all multi-index with , where .

Proof. Let . Suppose that

Set

It follows from proof of Theorem 6 that we have

Since , we see that

for all multi-index with . By Lemma 10, .
To prove the necessity, we suppose that . By Lemmas 1 and 11, for each multi-index and ,

Since

by Hölder’s inequality and Fubini’s theorem, we can obtain

It follows from Lemma 10 that we see that is bounded. This completes the proof.

4. Conclusions

In this paper, we characterize harmonic Bloch and Besov spaces by using the quantity . Since , our results can be viewed as the generalizations of Yoneda’s results (see [2]) into real harmonic setting. Furthermore, we obtain a characterization of the space in terms of , where .

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was partly supported by National Natural Science Foundation of China (Grant nos. 11501374, 11501284), Natural Science Foundation of Zhejiang Province (Grant no. LQ14A010006), and Natural Science Foundation of Hunan Province (Grant no. 2015JJ6095).

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